prime producing polynomial 2

[chris2be8]

Originally Posted by CRGreathouse

This is a good question.

Let f(x) be the putative prime-producing polynomial. gcd(x,1) = 1 for all x, so f(1) = p is prime. But then f(p + 1), f(2p + 1), ..., are all divisible by p, since each power of x will be 1 mod p. So they can be prime only if f(1) = f(p + 1) = f(2p + 1) = ..., which is possible only if f is the constant polynomial f(x) = p.

Thanks for that. I had said there could be a few other types of exception. But generalizing your argument:

Pick any n such that f(n)=p where p is prime. Then f(n+p), f(n+2p), etc will all be divisible by p since each power of x will be the same mod p as it is for f(n). So most values of n will generate a composite (unless it's a constant polynomial which can only generate 1 prime).

Chris

[ET_]

Originally Posted by paulunderwood

The quadratic x^2+1 has not been proven to generate an infinite number of primes. See 30 minutes into:
https://www.youtube.com/watch?v=rwH-5VhBPGc and if you have the time the whole video is worth watching.

Oh I see x is not an integer???

Wonderful video, thaank you

[MattcAnderson]

Hi Math People

Attached are slides for my write-up for n^2 + n + 41. I have removed all reference to the word bifurcation. That was incorrect. The points in the graph of divisors lie on parabolas.

Regards
Matt

[bhelmes]

A peaceful day for all members

there are some "new" results for the polynomial f(n)=n^2+n+41 for n<=2^35

http://www.devalco.de/basic_polynomi...a=1&b=1&c=41#7

For people who are not familiar with quadratic prime generators:

It is possible to make a sieving construction for quadratic irreducible polynomials
like f(n)=n² +1. The Sieve of Eratosthenes is a more specific variation of this construction. Normally the primes with p=f(n) or p|f(n) appear double periodically
on the polynomial. If you divide the appearing primes you can be sure that there only rest a prime or the number one.

Would nice to have a little feedback. The topic is quite interesting for people
who look for some prime generators.

An overview is in the web under
http://devalco.de/#106

Have a lot of fun with the primes
Bernhard

[MattcAnderson]

Hi Math People,

Thank you for considering my little project.

As I have state before the word "bifurcation" was incorrectly used by me to describe a graph.

New words are graph of discrete divisors.

For what it is worth.

Also, Bernhard, your work on seiving and quadratic functions is very interesting.

Regards,

Matthew

[MattcAnderson]

Hi all,

I continue to doodle with h(n). I define h(n) as n^2 + n + 41. We assume n is a positive integer. So far we know that no primes less than 40 ever divide h(n). Also, there are no positive integers n that make 59 divide h(n).

We prove these two facts by exhaustive search in a residue table.

Some of my progress is shown at this webpage -

https://sites.google.com/site/primeproducingpolynomial/

We know that if n is congruent to x mod y and (x,y) is on the curve

p(r,c) = (c*x – r*y)2 – r*(c*x – r*y) – x + 41*r^2 = 0

and 0<r<c, c>1 and gcd(r,c) = 1 and all four of r,c,x, and y are integers,

then h(n) is a composite number.

further, all n such that h(n) is a composite number are probably on p(r,c)=0.

Each such pair (r,c) yields integer points on a parabola.


Possible next steps -
We want to show that h(n) is prime an infinite number of times. p(r,c) is 0 for an infinite number of values x and y. Given that x and y are counting numbers and restrict x and y to be not a solution of p(r,c) = 0, can we infer that there are still an infinite set of pairs (x,y) such that n = x mod y will give us a prime value for h(n)?

This seems to be a hard question.

Regards,
Matt

[MattcAnderson]

Hi all,

There are two webpages that I have made regarding Prime Producing Polynomial.

The webpages are -

sites.google.com/site/mattc1anderson/prime-producing-polynomial

sites.google.com/site/primeproducingpolynomial

Regards,
Matt

[MattcAnderson]

Hi again all,

I added a counter example to a conjecture of mine that i about 2 years old.

Regarding our Prime Producing Polynomial Project for the enhancement of mathematical trivia database

Specifically,


counterexample to conjecture about x2_plus_x_plus_41.pdf

Cheers,

Matt

[MattcAnderson]

Hi all,

Today I coded a Maple worksheet and put it on the internet.
The file is called "an interesting graph with negatibes done.pdf".

You can click here to see.

Regards,
Matt